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Énergies renouvelables | Production éco-responsable | Transports innovants | Procédés éco-efficients | Ressources durables
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Comparison of numerical schemes for multiphase reactive transport
T. Faney, A. Michel, Q. L. Nguyen and L.Rouvray
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I. Introduction Motivation
References
Problem description
II. Formulation and Active Set Algorithm
III. Splitting Algorithm
IV. Results
V. Conclusion and Future Work
Outline
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Applications in Geosciences : CO2 Sequestration
Gas Production and Storage
EOR
Mineral Diagenesis
Other Applications : Batteries
Biological processes
I. Motivation
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I. Problem Description
System Sub-System Phase Species
Multi-Phase Multi-Species System
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I. Problem Description
Advection in Porous media
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I. Problem Description
Equilibrium chemical reactions For each reaction r
Closure Equations
Equilibrium reaction rates are unknown Rates elimination using chemical analysis
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Example System Water (H2O,OH-,H+,HCO3
-,Ca++) , Gas (CO2g), Calcite (CaCO3
s)
H2O = H+ + OH-
CO2g + H2O = HCO3- + H+
HCO3- + Ca++ = CaCO3
s + H+
Formula matrix A
13 equations (4 mass balance equations, 6 closure equations, 3 chemical equilibrium equations)
13 variables (P,𝝓𝒇, 𝝓𝒔,Sw,Sg,Ss,xH2O->CaCO3s)
I. Problem Description
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Discretization Global Implicit Approach (Implicit Euler)
Two-points finite volume scheme
Issue with phase appearance and disappearance Complementarity conditions
Active Set algorithm vs Semi-smooth Newton methods
New context variable I Ik = context of cell k
Equations and variables depend on Ik
Split active and non-active species, components and equations
II. Formulation and Active Set Algorithm
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Example continued Ik = {Gas, Calcite}
9 equations (4 mass balance equations, 5 closure equations, 0 chemical equilibrium equations) for cell k
9 variables (P,𝝓𝒇, 𝝓𝒔,Sg,Ss,xCO2g,xCaCO3s,NIH2O,NI
H+) for cell k
II. Formulation and Active Set Algorithm
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NA
NI
PA PI SI
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II. Formulation and Active Set Algorithm
Global non linear equation solver Each active variable is updated at the end of each Newton
iteration
Context variable also needs to be updated
Negative saturation phase disappearance
Equilibrium flash phase appearance
Equilibrium flash can be any 0D chemistry solver LMA / GEM formulation
External or Internal library
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III. Splitting Algorithm
Splitting occurs at different stages
Flow Reactive
Transport
Porosity
update
P, Sα njα
φ
tn tn+1
Transport Flash N P, Sα nj
α
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IV. Results – Active Set
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SHPCO2
case study
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IV. Results – Active Set
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IV. Results – Active Set
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IV. Results – Active Set
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IV. Results – Comparison
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SES Benchmark
Irina Sin and Jerome Corvisier, Ecole des Mines de Paris, Geosciences research center
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IV. Results – Comparison
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Simple benchmark
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IV. Results – Comparison
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T = 100 000 years
Active Set
Splitting (Non iterative)
Splitting (Iterative)
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IV. Conclusion and Future Work
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Active Set algorithm to model multi-phase multi-species flow in porous media
Comparison with splitting approach
Future work involves Non-Linear solver efficiency
Flash solver improvement
Semi-smooth Newton approach
Extension to kinetics and diffusion fluxes
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Thank you